Question
A charge $Q$ is placed at the centre of an imaginary hemispherical surface. Using symmetry arguments and the Gauss's law, find the flux of the electric field due to this charge through the surface of the hemisphere $($figure$).$
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Answer
From Guass's law, flux through a closed surface, $\phi=\frac{\text{Q}_{\text{en}}}{\epsilon_0},$
where $Q_{en} =$ charge enclosed by the closed surface Let us assume that a spherical closed surface in which the charge is enclosed is $Q.$ The flux through the sphere, $\phi=\frac{\text{Q}}{\epsilon_0}$
Hence for a hemisphere$($open bowl$)$, total flux through its curved surface,
$\phi'=\frac{\text{Q}}{\epsilon}\times\frac{1}{2}=\frac{\text{Q}}{2\epsilon_0}$
where $Q_{en} =$ charge enclosed by the closed surface Let us assume that a spherical closed surface in which the charge is enclosed is $Q.$ The flux through the sphere, $\phi=\frac{\text{Q}}{\epsilon_0}$
Hence for a hemisphere$($open bowl$)$, total flux through its curved surface,
$\phi'=\frac{\text{Q}}{\epsilon}\times\frac{1}{2}=\frac{\text{Q}}{2\epsilon_0}$
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